Three-valued semantics for hybrid MKNF knowledge bases revisited

Abstract

Knorr et al. (2011) [9] formulated a three-valued formalism for the logic of Minimal Knowledge and Negation as Failure (MKNF), and proposed a well-founded semantics for hybrid MKNF knowledge bases. The main results state that if a hybrid MKNF knowledge base has a three-valued MKNF model, its well-founded MKNF model exists, which is unique and can be computed by an alternating fixpoint construction. However, we show that these claims are erroneous. The goal of this paper is two-fold. First, we provide insight into why these claims fail to hold. This leads to a classification of hybrid MKNF knowledge bases into a hierarchy of four classes, of which the smallest one is what works for the well-founded semantics proposed by Knorr et al. After that, we characterize three-valued MKNF models into what we call stable partitions, which are similar to partial stable models in logic programming. The intuitive notion of stable partitions allows us to discover a new proof-theoretic method to determine whether a three-valued MKNF model exists for a given hybrid MKNF knowledge base. We can guess partial partitions and verify their stability by computing alternating fixpoints and by performing a consistency check. This algorithm can be applied to compute all three-valued MKNF models, as well as all two-valued ones originally formulated by Motik and Rosati for query answering. As a result, our work provides a uniform characterization of well-founded, two-valued, and all three-valued MKNF models, in terms of stable partitions and the alternating fixpoint construction.